Hamilton-Jacobi Theory and Canonical Transformations

In this chapter we want to present the basic features of the Hamilton—Jacobi theory, the centerpiece of analytical mechanics, which has played a major role in the development of the mathematical foundations of quantum mechanics as well as in the genesis of an analysis on manifolds. This theory is not only based on the fundamental work of Hamilton and Jacobi, but it also incorporates ideas of predecessors such as Fermat, Newton, Huygens and Johann Bernoulli among the old masters and Euler, Lagrange, Legendre, Monge, Pfaff, Poisson and Cauchy of the next generations. In addition the contributions of Lie, Poincaré and E. Cartan had a great influence on its final shaping.